Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ = [φ1, · · · , φr] be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ = [ψ1, · · · , ψr] an o.n. multiwavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r + 1)-dimensional o.n. scaling function vector φ := [φ>, φr+1] > and some corresponding o.n. multi-wavelet ψ are constructed in such a way that φ has p.p.o. = n > m and their two-scale symbols P ] andQ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r = 1, if we consider the m order Daubechies o.n. scaling function φm, then φ := [φm, φ2] > is a scaling function vector with p.p.o. > m. As another example, for r = 2, if we use the symmetric o.n. scaling function vector φ in our earlier work [3], then we obtain a new pair of scaling function vector φ = [φ>, φ3] > and multiwavelet ψ that not only increase the order of vanishing moments but also preserve symmetry.